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Автор: Grafiati

Опубліковано: 1 червня 2024

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1

Wang, Weifeng, Lei Yan, Junhao Hu, and Zhongkai Guo. "An Averaging Principle for Mckean–Vlasov-Type Caputo Fractional Stochastic Differential Equations." Journal of Mathematics 2021 (July 16, 2021): 1–11. http://dx.doi.org/10.1155/2021/8742330.

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In this paper, we want to establish an averaging principle for Mckean–Vlasov-type Caputo fractional stochastic differential equations with Brownian motion. Compared with the classic averaging condition for stochastic differential equation, we propose a new averaging condition and obtain the averaging convergence results for Mckean–Vlasov-type Caputo fractional stochastic differential equations.

2

Qiao, Huijie, and Jiang-Lun Wu. "Path independence of the additive functionals for McKean–Vlasov stochastic differential equations with jumps." Infinite Dimensional Analysis, Quantum Probability and Related Topics 24, no. 01 (March 2021): 2150006. http://dx.doi.org/10.1142/s0219025721500065.

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In this paper, the path independent property of additive functionals of McKean–Vlasov stochastic differential equations with jumps is characterized by nonlinear partial integro-differential equations involving [Formula: see text]-derivatives with respect to probability measures introduced by Lions. Our result extends the recent work16 by Ren and Wang where their concerned McKean–Vlasov stochastic differential equations are driven by Brownian motions.

3

Ma, Li, Fangfang Sun, and Xinfang Han. "Controlled Reflected McKean–Vlasov SDEs and Neumann Problem for Backward SPDEs." Mathematics 12, no. 7 (March 31, 2024): 1050. http://dx.doi.org/10.3390/math12071050.

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This paper is concerned with the stochastic optimal control problem of a 1-dimensional McKean–Vlasov stochastic differential equation (SDE) with reflection, of which the drift coefficient and diffusion coefficient can be both dependent on the state of the solution process along with its law and control. One backward stochastic partial differential equation (BSPDE) with the Neumann boundary condition can represent the value function of this control problem. Existence and uniqueness of the solution to the above equation are obtained. Finally, the optimal feedback control can be constructed by the BSPDE.

4

Narita, Kiyomasa. "The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field." Advances in Applied Probability 23, no. 2 (June 1991): 303–16. http://dx.doi.org/10.2307/1427750.

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The oscillator of the Liénard type with mean-field containing a large parameter α < 0 is considered. The solution of the two-dimensional stochastic differential equation with mean-field of the McKean type is taken as the response of the oscillator. By a rigorous evaluation of the upper bound of the displacement process depending on the parameter α, a one-dimensional limit diffusion process as α → ∞is derived and identified. Then our result extends the Smoluchowski–Kramers approximation for the Langevin equation without mean-field to the McKean equation with mean-field.

5

Narita, Kiyomasa. "The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field." Advances in Applied Probability 23, no. 02 (June 1991): 303–16. http://dx.doi.org/10.1017/s000186780002351x.

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Анотація:

The oscillator of the Liénard type with mean-field containing a large parameter α < 0 is considered. The solution of the two-dimensional stochastic differential equation with mean-field of the McKean type is taken as the response of the oscillator. By a rigorous evaluation of the upper bound of the displacement process depending on the parameter α, a one-dimensional limit diffusion process as α → ∞is derived and identified. Then our result extends the Smoluchowski–Kramers approximation for the Langevin equation without mean-field to the McKean equation with mean-field.

6

Pham, Huyên, and Xiaoli Wei. "Bellman equation and viscosity solutions for mean-field stochastic control problem." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 1 (January 2018): 437–61. http://dx.doi.org/10.1051/cocv/2017019.

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We consider the stochastic optimal control problem of McKean−Vlasov stochastic differential equation where the coefficients may depend upon the joint law of the state and control. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution of the process as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to probability measures recently introduced by [P.L. Lions, Cours au Collège de France: Théorie des jeux à champ moyens, audio conference 2006−2012], and a special Itô formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a verification theorem in our McKean−Vlasov framework. We give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. We also consider a notion of lifted viscosity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean−Vlasov control problem. Finally, we consider the case of McKean−Vlasov control problem with open-loop controls and discuss the associated dynamic programming equation that we compare with the case of closed-loop controls.

7

Bahlali, Khaled, Mohamed Amine Mezerdi, and Brahim Mezerdi. "Stability of McKean–Vlasov stochastic differential equations and applications." Stochastics and Dynamics 20, no. 01 (June 12, 2019): 2050007. http://dx.doi.org/10.1142/s0219493720500070.

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We consider McKean–Vlasov stochastic differential equations (MVSDEs), which are SDEs where the drift and diffusion coefficients depend not only on the state of the unknown process but also on its probability distribution. This type of SDEs was studied in statistical physics and represents the natural setting for stochastic mean-field games. We will first discuss questions of existence and uniqueness of solutions under an Osgood type condition improving the well-known Lipschitz case. Then, we derive various stability properties with respect to initial data, coefficients and driving processes, generalizing known results for classical SDEs. Finally, we establish a result on the approximation of the solution of a MVSDE associated to a relaxed control by the solutions of the same equation associated to strict controls. As a consequence, we show that the relaxed and strict control problems have the same value function. This last property improves known results proved for a special class of MVSDEs, where the dependence on the distribution was made via a linear functional.

8

Bao, Jianhai, Christoph Reisinger, Panpan Ren, and Wolfgang Stockinger. "First-order convergence of Milstein schemes for McKean–Vlasov equations and interacting particle systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2245 (January 2021): 20200258. http://dx.doi.org/10.1098/rspa.2020.0258.

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In this paper, we derive fully implementable first-order time-stepping schemes for McKean–Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean–Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.

9

Narita, Kiyomasa. "Asymptotic behavior of velocity process in the Smoluchowski–Kramers approximation for stochastic differential equations." Advances in Applied Probability 23, no. 2 (June 1991): 317–26. http://dx.doi.org/10.2307/1427751.

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Here a response of a non-linear oscillator of the Liénard type with a large parameter α ≥ 0 is formulated as a solution of a two-dimensional stochastic differential equation with mean-field of the McKean type. This solution is governed by a special form of the Fokker–Planck equation such as the Smoluchowski–Kramers equation, which is an equation of motion for distribution functions in position and velocity space describing the Brownian motion of particles in an external field. By a change of time and displacement we find that the velocity process converges to a one-dimensional Ornstein–Uhlenbeck process as α →∞.

10

Narita, Kiyomasa. "Asymptotic behavior of velocity process in the Smoluchowski–Kramers approximation for stochastic differential equations." Advances in Applied Probability 23, no. 02 (June 1991): 317–26. http://dx.doi.org/10.1017/s0001867800023521.

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Анотація:

Here a response of a non-linear oscillator of the Liénard type with a large parameter α ≥ 0 is formulated as a solution of a two-dimensional stochastic differential equation with mean-field of the McKean type. This solution is governed by a special form of the Fokker–Planck equation such as the Smoluchowski–Kramers equation, which is an equation of motion for distribution functions in position and velocity space describing the Brownian motion of particles in an external field. By a change of time and displacement we find that the velocity process converges to a one-dimensional Ornstein–Uhlenbeck process as α →∞.

11

Shen, Guangjun, Jie Xiang, and Jiang-Lun Wu. "Stochastic averaging principle for multi-valued McKean–Vlasov stochastic differential equations." Applied Mathematics Letters 141 (July 2023): 108629. http://dx.doi.org/10.1016/j.aml.2023.108629.

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12

Wen, Jianghui, Xiangjun Wang, Shuhua Mao, and Xinping Xiao. "Maximum likelihood estimation of McKean–Vlasov stochastic differential equation and its application." Applied Mathematics and Computation 274 (February 2016): 237–46. http://dx.doi.org/10.1016/j.amc.2015.11.019.

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13

Keck, David N., and Mark A. McKibben. "On a McKean‐Vlasov Stochastic Integro‐differential Evolution Equation of Sobolev‐Type." Stochastic Analysis and Applications 21, no. 5 (January 9, 2003): 1115–39. http://dx.doi.org/10.1081/sap-120024706.

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14

Lv, Li, Yanjie Zhang, and Zibo Wang. "Information upper bound for McKean–Vlasov stochastic differential equations." Chaos: An Interdisciplinary Journal of Nonlinear Science 31, no. 5 (May 2021): 051103. http://dx.doi.org/10.1063/5.0049874.

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15

Hutzenthaler, Martin, Thomas Kruse, and Tuan Anh Nguyen. "Multilevel Picard approximations for McKean-Vlasov stochastic differential equations." Journal of Mathematical Analysis and Applications 507, no. 1 (March 2022): 125761. http://dx.doi.org/10.1016/j.jmaa.2021.125761.

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16

Ahmed, N. U., and Xinhong Ding. "On invariant measures of nonlinear Markov processes." Journal of Applied Mathematics and Stochastic Analysis 6, no. 4 (January 1, 1993): 385–406. http://dx.doi.org/10.1155/s1048953393000310.

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17

Mehri, Sima, and Wilhelm Stannat. "Weak solutions to Vlasov–McKean equations under Lyapunov-type conditions." Stochastics and Dynamics 19, no. 06 (November 18, 2019): 1950042. http://dx.doi.org/10.1142/s0219493719500424.

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We present a Lyapunov-type approach to the problem of existence and uniqueness of general law-dependent stochastic differential equations. In the existing literature, most results concerning existence and uniqueness are obtained under regularity assumptions of the coefficients with respect to the Wasserstein distance. Some existence and uniqueness results for irregular coefficients have been obtained by considering the total variation distance. Here, we extend this approach to the control of the solution in some weighted total variation distance, that allows us now to derive a rather general weak uniqueness result, merely assuming measurability and certain integrability on the drift coefficient and some non-degeneracy on the dispersion coefficient. We also present an abstract weak existence result for the solution of law-dependent stochastic differential equations with merely measurable coefficients, based on an approximation with law-dependent stochastic differential equations with regular coefficients under Lyapunov-type assumptions.

18

Nie, Tianyang, and Ke Yan. "Extended mean-field control problem with partial observation." ESAIM: Control, Optimisation and Calculus of Variations 28 (2022): 17. http://dx.doi.org/10.1051/cocv/2022010.

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We study an extended mean-field control problem with partial observation, where the dynamic of the state is given by a forward-backward stochastic differential equation of McKean-Vlasov type. The cost functional, the state and the observation all depend on the joint distribution of the state and the control process. Our problem is motivated by the recent popular subject of mean-field games and related control problems of McKean-Vlasov type. We first establish a necessary condition in the form of Pontryagin’s maximum principle for optimality. Then a verification theorem is obtained for optimal control under some convex conditions of the Hamiltonian function. The results are also applied to studying linear-quadratic mean-filed control problem in the type of scalar interaction.

19

Mezerdi, Mohamed Amine. "Compactification in optimal control of McKean‐Vlasov stochastic differential equations." Optimal Control Applications and Methods 42, no. 4 (March 24, 2021): 1161–77. http://dx.doi.org/10.1002/oca.2721.

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20

Liu, Meiqi, and Huijie Qiao. "Parameter Estimation of Path-Dependent McKean-Vlasov Stochastic Differential Equations." Acta Mathematica Scientia 42, no. 3 (April 21, 2022): 876–86. http://dx.doi.org/10.1007/s10473-022-0304-8.

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21

Belomestny, Denis, and John Schoenmakers. "Projected Particle Methods for Solving McKean--Vlasov Stochastic Differential Equations." SIAM Journal on Numerical Analysis 56, no. 6 (January 2018): 3169–95. http://dx.doi.org/10.1137/17m1111024.

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22

Şen, Nevroz, and Peter E. Caines. "Nonlinear Filtering Theory for McKean--Vlasov Type Stochastic Differential Equations." SIAM Journal on Control and Optimization 54, no. 1 (January 2016): 153–74. http://dx.doi.org/10.1137/15m1013304.

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23

Carmona, René, and François Delarue. "Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics." Annals of Probability 43, no. 5 (September 2015): 2647–700. http://dx.doi.org/10.1214/14-aop946.

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24

Bencheikh, O., and B. Jourdain. "Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean." ESAIM: Proceedings and Surveys 65 (2019): 219–35. http://dx.doi.org/10.1051/proc/201965219.

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In this paper, we prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step h of a system of N interacting particles is 𝒪(N-1+h). We provide numerical experiments confirming this behaviour and showing that it extends to more general mean-field interaction and study the efficiency of the antithetic sampling technique on the same examples.

25

Narita, Kiyomasa. "Asymptotic analysis for interactive oscillators of the van der Pol type." Advances in Applied Probability 19, no. 1 (March 1987): 44–80. http://dx.doi.org/10.2307/1427373.

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We consider the N-oscillator system of the van der Pol type, which contains a small positive parameter ε multiplying the non-linear damping and the random disturbance. For a formulation of the output we take the solution X(t) = (Xi(t))i=1···,N of the system of 2N-dimensional stochastic differential equations. Rotating each component Xi(t) about the origin of the plane by an angle t, we find that on time scales of order 1/ε together with sufficiently large N each Xi(t) behaves as the equi-ultimately bounded solution of an equation of the McKean type admitting a stationary probability distribution.

26

Narita, Kiyomasa. "Asymptotic analysis for interactive oscillators of the van der Pol type." Advances in Applied Probability 19, no. 01 (March 1987): 44–80. http://dx.doi.org/10.1017/s0001867800016384.

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We consider the N-oscillator system of the van der Pol type, which contains a small positive parameter ε multiplying the non-linear damping and the random disturbance. For a formulation of the output we take the solution X(t) = (Xi (t)) i=1···, N of the system of 2N-dimensional stochastic differential equations. Rotating each component Xi (t) about the origin of the plane by an angle t, we find that on time scales of order 1/ε together with sufficiently large N each Xi (t) behaves as the equi-ultimately bounded solution of an equation of the McKean type admitting a stationary probability distribution.

27

Kotelenez, Peter M., and Thomas G. Kurtz. "Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type." Probability Theory and Related Fields 146, no. 1-2 (December 12, 2008): 189–222. http://dx.doi.org/10.1007/s00440-008-0188-0.

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28

Coppini, Fabio, Helge Dietert, and Giambattista Giacomin. "A law of large numbers and large deviations for interacting diffusions on Erdős–Rényi graphs." Stochastics and Dynamics 20, no. 02 (July 10, 2019): 2050010. http://dx.doi.org/10.1142/s0219493720500100.

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We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erdős–Rényi graph with parameter [Formula: see text], where [Formula: see text] is the size of the graph (i.e. the number of particles). If [Formula: see text], the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as [Formula: see text] to the solution of a PDE: a McKean–Vlasov (or Fokker–Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erdős–Rényi graphs with [Formula: see text], and properly rescaling the interaction to account for the dilution introduced by [Formula: see text]. However, these results have been proven under strong assumptions on the initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results — Law of Large Numbers and Large Deviation Principle — assuming only the convergence of the empirical measure of the initial condition.

29

Chen, Xingyuan, and Gonçalo dos Reis. "A flexible split‐step scheme for solving McKean‐Vlasov stochastic differential equations." Applied Mathematics and Computation 427 (August 2022): 127180. http://dx.doi.org/10.1016/j.amc.2022.127180.

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30

Chaudru de Raynal, P. E. "Strong well posedness of McKean–Vlasov stochastic differential equations with Hölder drift." Stochastic Processes and their Applications 130, no. 1 (January 2020): 79–107. http://dx.doi.org/10.1016/j.spa.2019.01.006.

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31

Wu, Fuke, Fubao Xi, and Chao Zhu. "On a class of McKean-Vlasov stochastic functional differential equations with applications." Journal of Differential Equations 371 (October 2023): 31–49. http://dx.doi.org/10.1016/j.jde.2023.06.022.

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32

Wen, Xueqi, Zhi Li, and Liping Xu. "Strong approximation of non-autonomous time-changed McKean–Vlasov stochastic differential equations." Communications in Nonlinear Science and Numerical Simulation 119 (May 2023): 107122. http://dx.doi.org/10.1016/j.cnsns.2023.107122.

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33

Mezerdi, Mohamed Amine, and Nabil Khelfallah. "Stability and prevalence of McKean–Vlasov stochastic differential equations with non-Lipschitz coefficients." Random Operators and Stochastic Equations 29, no. 1 (January 9, 2021): 67–78. http://dx.doi.org/10.1515/rose-2021-2053.

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Abstract We consider various approximation properties for systems driven by a McKean–Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover, the unique strong solutions may be constructed by an effective approximation procedure. Finally, we show that the set of bounded uniformly continuous coefficients for which the corresponding MVSDE have a unique strong solution is a set of second category in the sense of Baire.

34

Agarwal, A., S. De Marco, E. Gobet, J. G. López-Salas, F. Noubiagain, and A. Zhou. "Numerical approximations of McKean anticipative backward stochastic differential equations arising in initial margin requirements." ESAIM: Proceedings and Surveys 65 (2019): 1–26. http://dx.doi.org/10.1051/proc/201965001.

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We introduce a new class of anticipative backward stochastic differential equations with a dependence of McKean type on the law of the solution, that we name MKABSDE. We provide existence and uniqueness results in a general framework with relatively general regularity assumptions on the coefficients. We show how such stochastic equations arise within the modern paradigm of derivative pricing where a central counterparty (CCP) requires the members to deposit variation and initial margins to cover their exposure. In the case when the initial margin is proportional to the Conditional Value-at-Risk (CVaR) of the contract price, we apply our general result to define the price as a solution of a MKABSDE. We provide several linear and non-linear simpler approximations, which we solve using different numerical (deterministic and Monte-Carlo) methods.

35

Zhu, Min, and Yanyan Hu. "Least squares estimation for delay McKean–Vlasov stochastic differential equations and interacting particle systems." Communications in Mathematical Sciences 20, no. 1 (2022): 265–96. http://dx.doi.org/10.4310/cms.2022.v20.n1.a8.

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36

Chaudru de Raynal, P. E., and C. A. Garcia Trillos. "A cubature based algorithm to solve decoupled McKean–Vlasov forward–backward stochastic differential equations." Stochastic Processes and their Applications 125, no. 6 (June 2015): 2206–55. http://dx.doi.org/10.1016/j.spa.2014.11.018.

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37

Han, Jiequn, Ruimeng Hu, and Jihao Long. "Learning High-Dimensional McKean–Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence." SIAM Journal on Numerical Analysis 62, no. 1 (January 4, 2024): 1–24. http://dx.doi.org/10.1137/22m151861x.

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38

Wu, Dongxuan, Yaru Zhang, Liping Xu, and Zhi Li. "Strong convergence of Euler–Maruyama schemes for doubly perturbed McKean–Vlasov stochastic differential equations." Communications in Nonlinear Science and Numerical Simulation 132 (May 2024): 107927. http://dx.doi.org/10.1016/j.cnsns.2024.107927.

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39

Kotelenez, Peter. "A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation." Probability Theory and Related Fields 102, no. 2 (June 1995): 159–88. http://dx.doi.org/10.1007/bf01213387.

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40

Le Cavil, Anthony, Nadia Oudjane, and Francesco Russo. "Particle system algorithm and chaos propagation related to non-conservative McKean type stochastic differential equations." Stochastics and Partial Differential Equations: Analysis and Computations 5, no. 1 (August 29, 2016): 1–37. http://dx.doi.org/10.1007/s40072-016-0079-9.

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41

GÓMEZ-SERRANO, JAVIER, CARL GRAHAM, and JEAN-YVES LE BOUDEC. "THE BOUNDED CONFIDENCE MODEL OF OPINION DYNAMICS." Mathematical Models and Methods in Applied Sciences 22, no. 02 (February 2012): 1150007. http://dx.doi.org/10.1142/s0218202511500072.

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The bounded confidence model of opinion dynamics, introduced by Deffuant et al., is a stochastic model for the evolution of continuous-valued opinions within a finite group of peers. We prove that, as time goes to infinity, the opinions evolve globally into a random set of clusters too far apart to interact, and thereafter all opinions in every cluster converge to their barycenter. We then prove a mean-field limit result, propagation of chaos: as the number of peers goes to infinity in adequately started systems and time is rescaled accordingly, the opinion processes converge to i.i.d. nonlinear Markov (or McKean–Vlasov) processes; the limit opinion processes evolve as if under the influence of opinions drawn from its own instantaneous law, which are the unique solution of a nonlinear integro-differential equation of Kac type. This implies that the (random) empirical distribution processes converge to this (deterministic) solution. We then prove that, as time goes to infinity, this solution converges to a law concentrated on isolated opinions too far apart to interact, and identify sufficient conditions for the limit not to depend on the initial condition, and to be concentrated at a single opinion. Finally, we prove that if the equation has an initial condition with a density, then its solution has a density at all times, develop a numerical scheme for the corresponding functional equation, and show numerically that bifurcations may occur.

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Angiuli, Andrea, Christy V. Graves, Houzhi Li, Jean-François Chassagneux, François Delarue, and René Carmona. "Cemracs 2017: numerical probabilistic approach to MFG." ESAIM: Proceedings and Surveys 65 (2019): 84–113. http://dx.doi.org/10.1051/proc/201965084.

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This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [29], and by Huang, Caines, and Malhamé, see [26]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.

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CHI, Hongmei. "Multivalued stochastic McKean-Vlasov equation." Acta Mathematica Scientia 34, no. 6 (November 2014): 1731–40. http://dx.doi.org/10.1016/s0252-9602(14)60118-1.

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44

Hochgerner, Simon. "A Hamiltonian mean field system for the Navier–Stokes equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2218 (October 2018): 20180178. http://dx.doi.org/10.1098/rspa.2018.0178.

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We use a Hamiltonian interacting particle system to derive a stochastic mean field system whose McKean–Vlasov equation yields the incompressible Navier–Stokes equation. Since the system is Hamiltonian, the particle relabeling symmetry implies a Kelvin Circulation Theorem along stochastic Lagrangian paths. Moreover, issues of energy dissipation are discussed and the model is connected to other approaches in the literature.

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Kohlmann, M. "Stochastic differential equation." Metrika 33, no. 1 (December 1986): 246. http://dx.doi.org/10.1007/bf01894752.

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Hochgerner, Simon. "A Hamiltonian Interacting Particle System for Compressible Flow." Water 12, no. 8 (July 25, 2020): 2109. http://dx.doi.org/10.3390/w12082109.

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The decomposition of the energy of a compressible fluid parcel into slow (deterministic) and fast (stochastic) components is interpreted as a stochastic Hamiltonian interacting particle system (HIPS). It is shown that the McKean–Vlasov equation associated to the mean field limit yields the barotropic Navier–Stokes equation with density-dependent viscosity. Capillary forces can also be treated by this approach. Due to the Hamiltonian structure, the mean field system satisfies a Kelvin circulation theorem along stochastic Lagrangian paths.

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Ahmed, N. U., and X. Ding. "A semilinear Mckean-Vlasov stochastic evolution equation in Hilbert space." Stochastic Processes and their Applications 60, no. 1 (November 1995): 65–85. http://dx.doi.org/10.1016/0304-4149(95)00050-x.

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Cosso, Andrea, and Huyên Pham. "Zero-sum stochastic differential games of generalized McKean–Vlasov type." Journal de Mathématiques Pures et Appliquées 129 (September 2019): 180–212. http://dx.doi.org/10.1016/j.matpur.2018.12.005.

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49

Park, J. Y., P. Balasubramaniam, and Y. H. Kang. "Controllability of McKean–Vlasov Stochastic Integrodifferential Evolution Equation in Hilbert Spaces." Numerical Functional Analysis and Optimization 29, no. 11-12 (December 4, 2008): 1328–46. http://dx.doi.org/10.1080/01630560802580679.

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50

Keck, David N., and Mark A. McKibben. "Abstract semilinear stochastic Itó-Volterra integrodifferential equations." Journal of Applied Mathematics and Stochastic Analysis 2006 (July 4, 2006): 1–22. http://dx.doi.org/10.1155/jamsa/2006/45253.

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We consider a class of abstract semilinear stochastic Volterra integrodifferential equations in a real separable Hilbert space. The global existence and uniqueness of a mild solution, as well as a perturbation result, are established under the so-called Caratheodory growth conditions on the nonlinearities. An approximation result is then established, followed by an analogous result concerning a so-called McKean-Vlasov integrodifferential equation, and then a brief commentary on the extension of the main results to the time-dependent case. The paper ends with a discussion of some concrete examples to illustrate the abstract theory.